PROFESSOR: In this video, we're going to introduce statistical models.
Eventually, we're going to put it all together
and show you how statistical models can be very useful to analyze
high throughput experiments.
So we have already used modeling in previous videos.
We have used it, for example, when we used a t-distribution.
When we use a t-distribution, we're assuming
that our data is approximately normal.
So we had cases where you have 12 cases and 12 controls.
We compared them with a t-statistic.
We assumed that the t-statistic followed a t-distribution.
To do that, we see that we had to assume that these 12
values from each of the two groups came from a normal distribution.
In this figure here, we're showing you a check of that assumption,
but it is an assumption.
We don't know for sure.
We have 12 data points to check, but we don't know for sure if it's true.
So we assume that the data follow a normal distribution,
and then we can use the t-distribution approximation.
Now, there are other models that we use often in the life sciences.
For example, the binomial distribution is very commonly used.
Here I'm showing you the equation that tells us the probability of k successes
when we have N trials.
And the probability of success is p.
This formula is commonly used to model the number of heads
that we get when we toss N coins.
It fits that data very, very well.
But then we can also use it in life sciences examples.
For example, if we were to compute the probability of the number of Cytosines
in a random sequence of DNA, we could use a binomial distribution
to model that.
Also, if we have a next generation sequencing experiment,
where we're trying to decide if a specific location on the genome
it's a heterozygous or homozygous, if it is a heterozygous,
then the number of one genotype or the other that we see
should also follow a binomial distribution.
So it is used often in the life sciences.
Now, one thing that we have seen before is
that because the binomial distribution is modeling
the sum of independent random variables-- they're zero or one,
but they're random variables-- then if you divide by N,
you're basically looking at an average.
The central limit theorem then tells us that S
has to be approximately normal when N is large.
But this approximation is not appropriate
when p, the probability of success, is close to zero or one.
When that happens, then this brings up another distribution
that we commonly use in the life sciences, which
is the Poisson distribution.
To [? motivate ?] this, we're going to use the example of the lottery.
So suppose that we're trying to model the number of people
who win the lottery.
This is going to be s.
So if you think about it, this should follow a binomial distribution.
Let's go step by step.
So in the simulation that I've run here that produced the figure on the right,
we ran the lottery 1,000 times.
Each time we ran the lottery, there was a 1 in 10 million chance of winning.
But five million people-- this is N-- bought tickets.
Five million tickets were sold.
Now, s is a number of winners, the number of winning tickets.
This is a binomial.
We're adding up zeros and ones that have a very small probability,
but it's a positive probability, of being a success.
We run the simulation, and we notice that the data does not
follow a normal distribution.
It's not even close.
It's not even symmetric around a mean.
So we have a lot of zeros.
Many times nobody wins the lottery with this probability of winning.
The reason that the central limit theorem doesn't apply here
is because p is very close to zero.
There's another distribution that approximates this particular case,
and it's, as we said, the Poisson distribution.
Here's a formula for the Poisson distribution.
It tells us the probability of success, and here I'm
showing you the theoretical distribution that we would
obtain with lambda success rate of 0.5.
This is the expected number of winners, which is, in this case, 0.5
because we had five million tickets sold and a 1
in 10 million chance of winning.
You divide those two, and you get a rate of 1/2 winners per ticket sold.
You can see that the distribution is pretty well approximated,
at least in its shape, by the Poisson.